Are linear transformations of orthogonal vectors Orthogonal?

Question

Suppose there are 2 orthogonal vectors $v_1$ and $v_2$ so $v_1^Tv_2=0$. Suppose we take linear transformations of the two vectors. Let $A$ be the transformation matrix. Let $u_1=Av_1; u_2=Av_2;$ Are $u_1$ are $u_2$ orthogonal? i.e. is $u_1^Tu_2=0$? How to prove it. I tried the following way, but I'm stuck.

$u_1^Tu_2=0$
$(Av_1)^T(Av_2)=0$
$v_1^TA^TAv_2=0$

I'm stuck here, unable to proceed further.

If the above equation is not true for any random vectors $v_1$ and $v_2$, is it true for any specific vectors? What if $v_1$ and $v_2$ are in $Row$ $space$ $of$ $A$?

This holds true if either $v_1$ or $v_2$ is an eigen vector of $A^TA$ — Nagabhushan S N, Jun 07 '20 at 14:13

José Carlos Santos · Accepted Answer · 2018-05-19T08:19:58.153

1

It is not true in general, but it will be true if $A^TA=\lambda\operatorname{Id}$, for some scalar $\lambda$. In particular, it is true if $A$ is an orthogonal matrix ($\iff A^TA=\operatorname{Id}$).

edited May 19 '18 at 08:19

answered May 19 '18 at 08:13

José Carlos Santos

427,504

What if $v_1$ and $v_2$ are in row space of A? Is it true then? – Nagabhushan S N May 19 '18 at 08:17
@NagabhushanSN No, not even then. – José Carlos Santos May 19 '18 at 08:23
thank you. I was watching Video Lecture by Gilbert Strang on Singular Value Decomposition. In that he takes two orthonormal vectors $v_1$, $v_2$ in row-space of A. Let $\sigma_1$$u_1$=A$v_1$ and $\sigma_2$$u_2$=A$v_2$. He takes $u_1$ is orthogonal to $u_2$. How? – Nagabhushan S N May 19 '18 at 08:26
@NagabhushanSN Post this as another question, with a link to the video. – José Carlos Santos May 19 '18 at 08:29
here it is https://math.stackexchange.com/q/2777511/516816 – Nagabhushan S N May 19 '18 at 08:36
amswer link: https://math.stackexchange.com/a/2904688/516816 – Nagabhushan S N Dec 06 '22 at 14:48

user · Answer 2 · 2018-05-19T08:20:28.743

1

Not it is not true in general, just consider $e_1=(1,0)$, $e_2=(0,1)$ and

$$A=\begin{bmatrix}1&0\\1&1\end{bmatrix}$$

It is true when $A^TA=k\,I$, indeed

$$\langle Ae_i,Ae_j \rangle=e_i^TA^TAe_j=ke_i^Te_j=\langle e_i,e_j \rangle=0$$

edited May 19 '18 at 08:20

answered May 19 '18 at 08:15

user

154,566

Are linear transformations of orthogonal vectors Orthogonal?

2 Answers2